Question

For all $nϵN,7^{2n}-48n-1$ is divisible by.

• A. $25$
• B. $26$
• C. $1234$
• D. $2304$

Answer 1

Answer: (D)

$2304$

$7^{2n}-48n-1$

$=(49{)}^n-48n-1$
$=(1+48{)}^n-48n-1$

$=1+{}^n{C}_1\times 48+{}^n{C}_2\times {48}^2+{}^n{C}_3\times {48}^3+.....{+}^n{C}_n\times {48}^n-48n-1$

$=1+48n+{}^n{C}_2\times {48}^2+{}^n{C}_3\times {48}^3+.....{+}^n{C}_n\times {48}^n-48n-1$
$={}^n{C}_2\times {48}^2+{}^n{C}_3\times {48}^3+.....{+}^n{C}_n\times {48}^n$
$\Rightarrow 7^{2n}-48n-1$ is divisible by ${48}^2$, i.e., $2304$.